Question

5. Stoe-Geary preferences and Ramsey economy] Consider the standard Ramsey model of a closed economy, except that the representative house- hold's instantaneous utility function (felicity function) takes on the following Stone-Geary form, so that preferences are no n-homothetic u(c) 1-6 where č0 represents the subsistence level of per capita consumption. Suppose that the production function has the Cobb-Douglas form, and assume that there's no technological progress a. What is the intertemporal elasticity of substitution for the new form of the felicity function? Hint: Intertemporal elasticity of substitution, ơ, is cx u"(c) If >0, show how this elasticity measure changes as c rises.

Answer 1

The utility function is given as u(c) = (c - c bar)^1 - theta - 1/1 - theta Therefore u prime(c) = (1 - theta) (c - c bar)^- theta/1 - theta = (c - c bar)^- theta Therefore u doubleprime(c) = -theta(c - c bar)^-theta - 1 Therefore sigma = u prime(c)/c times u doubleprime(c) = -(c - c bar)^- theta/- theta (c - c bar)^- (theta + 1) times c = (c - c bar)^- theta/theta c (c - c bar)^- (theta + 1) = c - c bar/theta c Therefore sigma = 1/theta dot c bar/theta c Therefore partial differential sigma/partial differential c = c v/theta c^2 > 0 [sigma rises if c rises sigma falls if c falls]